Polynesians may have beat computers to using binary

A mixed binary/decimal system may have made some calculations easier.

The decimal system of counting is part of our language, math, and the measurement units used by all right-thinking nations. It's so deeply engrained in how we operate that it's often difficult to imagine using anything else. However, it's mostly a historic accident, based on the number of fingers we happen to have. Although the vast majority of societies used decimal numbers, some developed systems based on five or 20 digits instead. But there were also some rare exceptions. A new paper in PNAS performs an analysis of a Polynesian culture's language, and it concludes that its speakers developed a mixed decimal/binary system. The researchers then go on to argue that the inclusion of binary made certain math operations much easier.

There are some instances of binary systems being used. For example, the paper cites a language from Papua New Guinea that only includes words for one and two. But it's not a full binary system in that there's no concept of larger digits; these are simply represented by additive combinations of the two digits (so five is expressed as "2 + 2 + 1").

In contrast, the authors argue that the indigenous people of Mangareva performed full binary calculations but layered the results on top of a decimal counting system. The trick is that there is nobody left who actually uses the Mangarevan numerical system; instead, it has to be inferred from the language and cultural background of the people.

Mangareva was settled during the Polynesian expansion. It's a small group of islands at the eastern edge of French Polynesia where things start to thin out before remote outposts like Pitcairn Island and Easter Island. Nevertheless, it was fully incorporated into the Polynesian trade network, which meant that its residents had to be able to keep track of trade goods. In addition, many of these goods (like sea turtles and fish) were used by political figures as gestures of munificence during formal feasts with their subjects. Keeping track of just how generous these gifts were was an important part of the political culture.

So counting large numbers was a critical part of the Mangarevan culture. Polynesians as a whole used a decimal system of counting, but different island groups often had distinct terms for different groupings of 10. The researchers describe the Mangarevan language's specific terms for groups of 10 and then show how these could be used as a form of binary, allowing calculations to rapidly manipulate groups of 10 to conveniently perform addition, subtraction, multiplication, and division. The system, they argue, would allow large groups of trade goods to be rapidly inventoried with a relatively small cognitive load, essential for a culture without any writing.

(To complicate matters a bit further, different types of goods were handled in batches of different numbers. For example, you always counted individual sea turtles, but fish were handled in groups of two. If you said there were 40 fish, you actually had 80. But the math was all done by counting groups of two.)

The authors were able to make all these inferences by examining the Mangarevan language, which is still spoken by roughly 1,000 people in the islands. However, the actual math system has been lost, replaced by a full decimal system introduced by French missionaries. The best we can currently do is infer that it would have been easier to handle some things in binary; we can't confirm that this was the mental process used by the Mangarevans before the arrival of Europeans.

If right, however, the Polynesians were using a binary system for a few hundred years before Leibniz introduced it to European thought around 1700. And it was still a few centuries before electronics made binary a central part of most calculations.

Now all we need to do is find an ancient society that used hexadecimal.

If I'm not recalling incorrectly, the ancient sumarians? babylonians? Had a base 60 counting system (and the origin of having 360 degrees in a circle... conveniently evenly divisible by 60).

Yes, the Babylonians used base 60 (sexagesimal), which they inherited from the Sumerians and Akkadians. It's really a more convenient base than 10, as it's divisible by 1, 2, 3, 4, 5, 6, 10, 12, 15, 20, and 30, while 10 is only divisible by 1, 2, and 5. It makes fractions easier.

Now all we need to do is find an ancient society that used hexadecimal.

If I'm not recalling incorrectly, the ancient sumarians? babylonians? Had a base 60 counting system (and the origin of having 360 degrees in a circle... conveniently evenly divisible by 60).

Sexagesimal makes a lot of sense, particularly for a primitive society. They would count by using one thumb to count each segment on that hand, then count one finger on the other hand for each base unit (12) to count to 60 on their hands. It also divides nicely by a whole lot of other numbers.

The weird part comes from the fact that they also used base5, 10, 12, 20, and 100. Intermixedly, and sometimes with words instead of symbols. So it'd be kinda like seeing "3 + 0xC plus 0o15 = XXVIII".

Now all we need to do is find an ancient society that used hexadecimal.

If I'm not recalling incorrectly, the ancient sumarians? babylonians? Had a base 60 counting system (and the origin of having 360 degrees in a circle... conveniently evenly divisible by 60).

Yes, the Babylonians used base 60 (sexagesimal), which they inherited from the Sumerians and Akkadians. It's really a more convenient base than 10, as it's divisible by 1, 2, 3, 4, 5, 6, 10, 12, 15, 20, and 30, while 10 is only divisible by 1, 2, and 5. It makes fractions easier.

And it's easy to count on your fingers, using your thumb to trace the 12 finger joints on one hand and using each of the fingers and thumb on your other hand to track the multiples "column".

Now all we need to do is find an ancient society that used hexadecimal.

If I'm not recalling incorrectly, the ancient sumarians? babylonians? Had a base 60 counting system (and the origin of having 360 degrees in a circle... conveniently evenly divisible by 60).

Yes, the Babylonians used base 60 (sexagesimal), which they inherited from the Sumerians and Akkadians. It's really a more convenient base than 10, as it's divisible by 1, 2, 3, 4, 5, 6, 10, 12, 15, 20, and 30, while 10 is only divisible by 1, 2, and 5. It makes fractions easier.

And it's easy to count on your fingers, using your thumb to trace the 12 finger joints on one hand and using each of the fingers and thumb on your other hand to track the multiples "column".

Now all we need to do is find an ancient society that used hexadecimal.

If I'm not recalling incorrectly, the ancient sumarians? babylonians? Had a base 60 counting system (and the origin of having 360 degrees in a circle... conveniently evenly divisible by 60).

Yes, the Babylonians used base 60 (sexagesimal), which they inherited from the Sumerians and Akkadians. It's really a more convenient base than 10, as it's divisible by 1, 2, 3, 4, 5, 6, 10, 12, 15, 20, and 30, while 10 is only divisible by 1, 2, and 5. It makes fractions easier.

I'm not sure what type of glyphs they were using for digits, but if they had to have 60 unique ones, that's quite a bit of overhead. I would've gone with 12 (1,2,3,4,6), which would also make the clock make more sense. Also, that makes 60, "50" which is just interesting and a circle "260" degrees -- at least i think my quick math on that is right ;- ).

Now all we need to do is find an ancient society that used hexadecimal.

If I'm not recalling incorrectly, the ancient sumarians? babylonians? Had a base 60 counting system (and the origin of having 360 degrees in a circle... conveniently evenly divisible by 60).

Yes, the Babylonians used base 60 (sexagesimal), which they inherited from the Sumerians and Akkadians. It's really a more convenient base than 10, as it's divisible by 1, 2, 3, 4, 5, 6, 10, 12, 15, 20, and 30, while 10 is only divisible by 1, 2, and 5. It makes fractions easier.

I'm not sure what type of glyphs they were using for digits, but if they had to have 60 unique ones, that's quite a bit of overhead. I would've gone with 12 (1,2,3,4,6), which would also make the clock make more sense. Also, that makes 60, "50" which is just interesting and a circle "300" degrees -- at least i think my quick math on that is right ;- ).

They actually only had two symbols, and here it gets a bit weird. One symbol was for units. The other was for 10s. So, they also used as decimal system internal to the sexagesimal system.

They worked like Roman numerals, kind of. If I was their unit symbol, and X the tens, XII would be 12. Or maybe 12x60, or 12/60. Context mattered.

Now all we need to do is find an ancient society that used hexadecimal.

If I'm not recalling incorrectly, the ancient sumarians? babylonians? Had a base 60 counting system (and the origin of having 360 degrees in a circle... conveniently evenly divisible by 60).

Yes, the Babylonians used base 60 (sexagesimal), which they inherited from the Sumerians and Akkadians. It's really a more convenient base than 10, as it's divisible by 1, 2, 3, 4, 5, 6, 10, 12, 15, 20, and 30, while 10 is only divisible by 1, 2, and 5. It makes fractions easier.

I'm not sure what type of glyphs they were using for digits, but if they had to have 60 unique ones, that's quite a bit of overhead. I would've gone with 12 (1,2,3,4,6), which would also make the clock make more sense. Also, that makes 60, "50" which is just interesting and a circle "300" degrees -- at least i think my quick math on that is right ;- ).

Sumerian and by extension Babylonian was written using a wedge shaped stylus pressed into clay. The digits themselves are represented decimally. Two angled marks, <, represented 10 and a straight mark, |, represented 1. So, the digit 33 would be written similar to <<<|||. Larger single units would be grouped together to make the digit more compact e.g. 9 | would be written as 3 rows of 3 smaller marks.

The Sumerians used written language mostly/originally for trading and inventory purposes and so being able to tell a number at a glance was paramount. I doubt that lots of greater than 60 were frequently traded so the written numeral system may have been functionally decimal.

Now all we need to do is find an ancient society that used hexadecimal.

There are some aboriginal American cultures that used to count in octal. They were strong weaving cultures, so they counted based on how many strands of yarn they could hold between their fingers.

Given the prevalence of computers and the importance of binary, I'd like to see a greater emphasis for either octal or hexadecimal in primary schools. I know octal isn't used as much anymore, but its smaller size has the advantage of being within the "Magical" range of "7 plus or minus 2"-- and the hand thing.

And it's easy to count on your fingers, using your thumb to trace the 12 finger joints on one hand and using each of the fingers and thumb on your other hand to track the multiples "column".

Actually, trying it you can throw in the finger tips and get to 16*16 (+ another 16 technically, before you run out) - hexadecimal!

I still use "finger math" counting sometimes to get to 59 on my hands (right fingers 1's, thumb for 5, left fingers/thumbs 10's), might have to give that a go to get to 272. I suspect my brain will struggle though, the 10-base of finger math makes it easier.

Silly question: But given that there is no written record of the language, isn't it slightly more probably that elements of their spoken language adapted in response to the introduction of a binary system that has now superseded the original system?

Whilst the history explaining how useful such a system would be to them is interesting, it doesn't necessitate it. And as far as I can see, there's no actual evidence in support of it?

Aren't the primitive Americans still using the "Imperial units system" based on duodecimal or some arbitrary number system, even when Rest of the World™ is using modern metric system based on decimal number system? Even the "imperial state" where the Imperial system was invented already converted to metric system.It is no wonder that any seemingly strange number system may be used by any society or group, whatever.

the paper cites a language from Papua New Guinea that only includes words for one and two. But it's not a full binary system in that there's no concept of larger digits; these are simply represented by additive combinations of the two digits (so five is expressed as "2 + 2 + 1").

That's kind of odd thing to say. I mean, by that reasoning, we don't use a full decimal system because our larger digits are also additive combinations of smaller digits. For example, the word for 13 is a contraction of "three and ten"; the word for 75 is a contraction of "seventy and five", etc. etc. Some of these original constructions are still preserved in old sayings and folk songs ("Four and twenty blackbirds baked in a pie...").

Quote:

To complicate matters a bit further, different types of goods were handled in batches of different numbers. For example, you always counted individual sea turtles, but fish were handled in groups of two. If you said there were 40 fish, you actually had 80. But the math was all done by counting groups of two.

We still do things like this in modern English when it comes to paired items. We also do the reverse, e.g. "10 pairs of pants" = 10 items, not 20.

Aren't the primitive Americans still using the "Imperial units system" based on duodecimal or some arbitrary number system, even when Rest of the World™ is using modern metric system based on decimal number system?

Decimal is no less arbitrary, and although I was half-kidding before duodecimal actually has a lot going for it in terms of divisibility and alignment with naturally occurring phenomena. The reason most cultures don't use a decimal time system isn't just inertia.

The thing about our numerical systems, is that if it was truly based on how many fingers (incl. thumbs) we have, then it would be a base-11 system, not base-10, because we now always start with 0 - (decimal is 0-9, not 1-10). Now, I know that not every numerical system we've invented over the years started with 0 - but all the modern systems should do so - therefore it should be possible for an old base-10 system to now be recognised and understood as base-11, if it started with 1, not 0.

Aren't the primitive Americans still using the "Imperial units system" based on duodecimal or some arbitrary number system, even when Rest of the World™ is using modern metric system based on decimal number system? Even the "imperial state" where the Imperial system was invented already converted to metric system.It is no wonder that any seemingly strange number system may be used by any society or group, whatever.

Stepping over the obvious troll, Britain uses both the Imperial (known as "English" in the US) system and the metric system. The later is slowly but surely edging out the former, but Imperial is still officially used for certain things, e.g. distance and speed on the road.

Also, does primitive Dr.Appleseed still use 360 degrees for a circle and 24 hours for a day? How arbitrary! Clearly you should switch to decimal everywhere.

When I was a kid in England we used a mixed decimal/vigesimal/duodecimal system. And since it was for money, we had to be good at it:12 pennies in a shilling20 shillings in a poundCount the pounds in decimal from there.And that's not even taking into account the ha'penny coin (half penny), or the guinea (21 shillings).

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